3.860 \(\int \sqrt{d+e x} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=119 \[ -\frac{2 \left (c d^2-c e^2 x^2\right )^{5/2}}{9 c e \sqrt{d+e x}}-\frac{16 d \left (c d^2-c e^2 x^2\right )^{5/2}}{63 c e (d+e x)^{3/2}}-\frac{64 d^2 \left (c d^2-c e^2 x^2\right )^{5/2}}{315 c e (d+e x)^{5/2}} \]

[Out]

(-64*d^2*(c*d^2 - c*e^2*x^2)^(5/2))/(315*c*e*(d + e*x)^(5/2)) - (16*d*(c*d^2 - c
*e^2*x^2)^(5/2))/(63*c*e*(d + e*x)^(3/2)) - (2*(c*d^2 - c*e^2*x^2)^(5/2))/(9*c*e
*Sqrt[d + e*x])

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Rubi [A]  time = 0.163171, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ -\frac{2 \left (c d^2-c e^2 x^2\right )^{5/2}}{9 c e \sqrt{d+e x}}-\frac{16 d \left (c d^2-c e^2 x^2\right )^{5/2}}{63 c e (d+e x)^{3/2}}-\frac{64 d^2 \left (c d^2-c e^2 x^2\right )^{5/2}}{315 c e (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[d + e*x]*(c*d^2 - c*e^2*x^2)^(3/2),x]

[Out]

(-64*d^2*(c*d^2 - c*e^2*x^2)^(5/2))/(315*c*e*(d + e*x)^(5/2)) - (16*d*(c*d^2 - c
*e^2*x^2)^(5/2))/(63*c*e*(d + e*x)^(3/2)) - (2*(c*d^2 - c*e^2*x^2)^(5/2))/(9*c*e
*Sqrt[d + e*x])

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Rubi in Sympy [A]  time = 15.1976, size = 102, normalized size = 0.86 \[ - \frac{64 d^{2} \left (c d^{2} - c e^{2} x^{2}\right )^{\frac{5}{2}}}{315 c e \left (d + e x\right )^{\frac{5}{2}}} - \frac{16 d \left (c d^{2} - c e^{2} x^{2}\right )^{\frac{5}{2}}}{63 c e \left (d + e x\right )^{\frac{3}{2}}} - \frac{2 \left (c d^{2} - c e^{2} x^{2}\right )^{\frac{5}{2}}}{9 c e \sqrt{d + e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(1/2)*(-c*e**2*x**2+c*d**2)**(3/2),x)

[Out]

-64*d**2*(c*d**2 - c*e**2*x**2)**(5/2)/(315*c*e*(d + e*x)**(5/2)) - 16*d*(c*d**2
 - c*e**2*x**2)**(5/2)/(63*c*e*(d + e*x)**(3/2)) - 2*(c*d**2 - c*e**2*x**2)**(5/
2)/(9*c*e*sqrt(d + e*x))

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Mathematica [A]  time = 0.0505042, size = 62, normalized size = 0.52 \[ -\frac{2 c (d-e x)^2 \left (107 d^2+110 d e x+35 e^2 x^2\right ) \sqrt{c \left (d^2-e^2 x^2\right )}}{315 e \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[d + e*x]*(c*d^2 - c*e^2*x^2)^(3/2),x]

[Out]

(-2*c*(d - e*x)^2*Sqrt[c*(d^2 - e^2*x^2)]*(107*d^2 + 110*d*e*x + 35*e^2*x^2))/(3
15*e*Sqrt[d + e*x])

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Maple [A]  time = 0.012, size = 55, normalized size = 0.5 \[ -{\frac{ \left ( -2\,ex+2\,d \right ) \left ( 35\,{e}^{2}{x}^{2}+110\,dex+107\,{d}^{2} \right ) }{315\,e} \left ( -c{e}^{2}{x}^{2}+c{d}^{2} \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(1/2)*(-c*e^2*x^2+c*d^2)^(3/2),x)

[Out]

-2/315*(-e*x+d)*(35*e^2*x^2+110*d*e*x+107*d^2)*(-c*e^2*x^2+c*d^2)^(3/2)/e/(e*x+d
)^(3/2)

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Maxima [A]  time = 0.738163, size = 111, normalized size = 0.93 \[ -\frac{2 \,{\left (35 \, c^{\frac{3}{2}} e^{4} x^{4} + 40 \, c^{\frac{3}{2}} d e^{3} x^{3} - 78 \, c^{\frac{3}{2}} d^{2} e^{2} x^{2} - 104 \, c^{\frac{3}{2}} d^{3} e x + 107 \, c^{\frac{3}{2}} d^{4}\right )}{\left (e x + d\right )} \sqrt{-e x + d}}{315 \,{\left (e^{2} x + d e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-c*e^2*x^2 + c*d^2)^(3/2)*sqrt(e*x + d),x, algorithm="maxima")

[Out]

-2/315*(35*c^(3/2)*e^4*x^4 + 40*c^(3/2)*d*e^3*x^3 - 78*c^(3/2)*d^2*e^2*x^2 - 104
*c^(3/2)*d^3*e*x + 107*c^(3/2)*d^4)*(e*x + d)*sqrt(-e*x + d)/(e^2*x + d*e)

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Fricas [A]  time = 0.212991, size = 153, normalized size = 1.29 \[ \frac{2 \,{\left (35 \, c^{2} e^{6} x^{6} + 40 \, c^{2} d e^{5} x^{5} - 113 \, c^{2} d^{2} e^{4} x^{4} - 144 \, c^{2} d^{3} e^{3} x^{3} + 185 \, c^{2} d^{4} e^{2} x^{2} + 104 \, c^{2} d^{5} e x - 107 \, c^{2} d^{6}\right )}}{315 \, \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-c*e^2*x^2 + c*d^2)^(3/2)*sqrt(e*x + d),x, algorithm="fricas")

[Out]

2/315*(35*c^2*e^6*x^6 + 40*c^2*d*e^5*x^5 - 113*c^2*d^2*e^4*x^4 - 144*c^2*d^3*e^3
*x^3 + 185*c^2*d^4*e^2*x^2 + 104*c^2*d^5*e*x - 107*c^2*d^6)/(sqrt(-c*e^2*x^2 + c
*d^2)*sqrt(e*x + d)*e)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (- c \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{3}{2}} \sqrt{d + e x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**(1/2)*(-c*e**2*x**2+c*d**2)**(3/2),x)

[Out]

Integral((-c*(-d + e*x)*(d + e*x))**(3/2)*sqrt(d + e*x), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (-c e^{2} x^{2} + c d^{2}\right )}^{\frac{3}{2}} \sqrt{e x + d}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-c*e^2*x^2 + c*d^2)^(3/2)*sqrt(e*x + d),x, algorithm="giac")

[Out]

integrate((-c*e^2*x^2 + c*d^2)^(3/2)*sqrt(e*x + d), x)